What is ratio of Bohr magneton to the nuclear magneton?

Options

A. `m_(1)//m_(e)`
B. `m_(p)^(2)//m_(e)^(2)`
C. 1
D. `(m_(e))/(m_(p))`

Explanation

To find the ratio of Bohr magneton (\( \mu_B \)) to nuclear magneton (\( \mu_N \)), we start by defining both quantities.

1. **Definition of Bohr Magneton (\( \mu_B \))**:
\[ \mu_B = \frac{e \hbar}{2 m_e} \]
where:
- \( e \) = charge of the electron
- \( \hbar \) = reduced Planck's constant
- \( m_e \) = mass of the electron

2. **Definition of Nuclear Magneton (\( \mu_N \))**:
\[ \mu_N = \frac{e \hbar}{2 m_N} \]
where:
- \( m_N \) = mass of the nucleon (proton or neutron, typically taken as the mass of a proton for this calculation)

3. **Finding the Ratio \( \frac{\mu_B}{\mu_N} \)**:
\[ \frac{\mu_B}{\mu_N} = \frac{\frac{e \hbar}{2 m_e}}{\frac{e \hbar}{2 m_N}} \]
Simplifying this expression:
\[ \frac{\mu_B}{\mu_N} = \frac{e \hbar}{2 m_e} \cdot \frac{2 m_N}{e \hbar} \]
The \( e \hbar \) and \( 2 \) cancel out:
\[ \frac{\mu_B}{\mu_N} = \frac{m_N}{m_e} \]

4. **Substituting Mass Values**:
The mass of a proton (\( m_p \)) is approximately 1836 times the mass of an electron (\( m_e \)):
\[ m_N \approx m_p \approx 1836 m_e \]
Therefore:
\[ \frac{\mu_B}{\mu_N} = \frac{m_p}{m_e} \]

5. **Final Result**:
Thus, the ratio of Bohr magneton to nuclear magneton is:
\[ \frac{\mu_B}{\mu_N} = \frac{m_p}{m_e} \]